Answer
$k=\dfrac{-17}{12}$
Work Step by Step
The factor theorem states that when $f(a)=0$, then we have $(x-a)$ as a factor of $f(x)$ and when $(x-a)$ is a factor of $f(x)$, then $f(a)=0$.
As per the given equation, when $f(x)=x^4=kx^3+kx^2+1$ has a factor $x+2$, then by the factor theorem $f(-2)=0$.
We simplify the given equation as follows:
$f(-2)=(-2)^4-k(-2)^3+(-2)^2k+1=0 \\ 16+8k+4k+1=0\\ -12k =17 \\ k=\dfrac{-17}{12}$
